JEE Advance - Physics (2011 - Paper 2 Offline - No. 14)

A series RC-current is connected to AC voltage source. Consider two cases : (A) When C is without a dielectric medium and (B) when C is filled with dielectric of constant 4. The current IR through the resistor and voltage VC across the capacitor are compared in the two cases. Which of the following is/are true?
$$I_R^A > I_R^B$$
$$I_R^A < I_R^B$$
$$V_C^A > V_C^B$$
$$V_C^A < V_C^B$$

Explanation

In case A,

The capacitive reactance is

$$X_C^A = {1 \over {\omega C}}$$

Impedance of the circuit is

$${Z_A} = \sqrt {{{(R)}^2} + {{\left( {{1 \over {\omega C}}} \right)}^2}} $$

$$I_R^A = {V \over {\sqrt {{{(R)}^2} + {{\left( {{1 \over {\omega C}}} \right)}^2}} }}$$ ...... (i)

$$V_C^A = {{I_R^A} \over {\omega C}} = {V \over {\sqrt {{{(R\omega C)}^2} + 1} }}$$ ...... (ii)

In case B,

The capacitive reactance is

$$X_C^B = {1 \over {\omega (4C)}} = {1 \over {4\omega C}}$$

Impedance of the circuit is

$${Z_B} = \sqrt {{R^2} + {{\left( {{1 \over {4\omega C}}} \right)}^2}} $$

$$I_R^B = {V \over {\sqrt {{R^2} + {{\left( {{1 \over {4\omega C}}} \right)}^2}} }}$$ ..... (iii)

$$V_C^B = {V \over {\sqrt {{{(4R\omega C)}^2} + 1} }}$$ ..... (iv)

From (i) and (iii), we conclude that

$$I_R^A < I_R^B$$

From (ii) and (iv), we conclude that

$$V_C^A > V_C^B$$

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